The increasing complexity and uncertainty surrounding the development of embedded applications continues to escalate their development costs and to challenge their window of opportunity. Model-based design (MBD) has emerged as a methodology capable of addressing these difficulties inherent in designing modern embedded and control systems. Ensuring that a discrete-time model (DTM) can be processed efficiently and keeping the accuracy and stability of the original continuous-time model (CTM) in predefined limits imposes a burdensome constraint upon the design of the system algorithms and hardware architecture.
Techniques for finding the discrete-time representation of a CTM typically fall into one of three major classes. The first and perhaps the simplest methodology is known as the matched pole-zero approximation method. It is based on the idea that when applying the Z-transform on a given system, then poles of the continuous system (wp) and those of the discrete equivalent (zp) are related via the exponential relation zp=e{wp}. Unfortunately, such an exponential relation is not accurate enough to map the zeros of the continuous system. To overcome this problem, the matched pole-zero method relies on a series of heuristic rules based on the relation z=e{sT} and adjusting the gain of the discrete system to match the low frequency gain of the continuous system. A summary of those heuristic rules can be widely found in the literature. The second approach, known as Hold Equivalent Models, seeks to approximate a CTM with transfer function H(s) through sampling the input, e(t), at discrete time points, t=nT, n=0, 1, 2, . . . , holding the samples through some means in the interval nT<=t<=(n+1)T, using the output of the holder to drive the CTM, and sampling its output at the same discrete time instants. The notion of holding the sample is typically carried out through a set of approximating polynomials, where, the approximating polynomials could have orders equal to zero or one leading to hold equivalents that are called zero-order (ZOH) and first-order hold (FOH). Although higher-order holders are possible, they are not normally used in practical applications because it can be difficult to obtain, expensive and result in lower performance. The third approach known as the s-to-z Mapping Models is typically constructed by considering a numerical solver of a system of differential equations. The solver is typically used to approximate the exact solution, denoted by x(t), of differential equations at discrete time instants, i.e., t=tn, n=1, 2, 3 . . . , through its previously known past points on the solution, i.e. at t=tm (m<n), its derivatives at past points or combinations thereof.
Stability and accuracy are two important issues that must be addressed in any approach used to construct a DTM in any approach. The accuracy of a DTM is usually expressed by the order of the rule approximating the CTM. This order is an integer number, where higher-order typically means higher accuracy. Stability, on the other hand loosely speaking, refers to the notion of having stable poles for the DTM whenever its CTM counterpart is stable with stable poles. A desirable DTM is therefore a stable model with the highest possible order. More specifically, the key challenge to develop modern high accuracy embedded applications is not simply generating high-order DTM approximations, but also guaranteeing their unconditional stability.
A particular problem with the construction of the DTM based on the third approach is that the numerical solver usually used for that purpose has an inherent conflict between stability and order. In other words, the stability regions decrease with increasing order. Thus a DTM with high-order is more likely to be unstable, and thereby defeating the purpose of the high-order. Indeed, it has been proven that numerical solvers based on general integration formulas (IF) such as Linear Multi Step (LMS) methods cannot be unconditionally stable (or A-stable) if its order increases beyond order 2. As a result, all s-to-z mapping techniques derived from LMS-IF formulae will inherit this conflict making it impossible to obtain unconditionally stable DTM approximations of a CTM of orders higher than 2. Given that embedded systems are required to be faster and with more complex structure, the need for high speed and high accuracy discretization methods will be exacerbated.
It is well-known that the goal of embedding a DTM model of a CTM in hardware is fraught with many difficulties, such as                1. Embedded processors have limited resources as well as limited speed and power requirements.        2. Practical realization of a DTM in hardware such as a signal processing circuit should not incur a faster-than-linear growth in memory or processing resources, given the limited resources on these circuits mentioned above.        3. Accuracy and stability are two requirements that are indispensable and that must be preserved in the DTM model. However, as noted earlier, those two requirements are always in conflict, making that one must be always obtained at the expense of the other.        
Therefore there is a need for an improved system, method and apparatus for mapping a continuous-time model (CTM) into a discrete-time model (DTM).